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NUMERICAL ENTROPY PRODUCTION AS SMOOTHNESS INDICATOR FOR SHALLOW WATER EQUATIONS

Part of:Basic methods in fluid mechanicsHyperbolic equations and systemsPartial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  28 November 2019

SUDI MUNGKASIOpen the ORCID record for SUDI MUNGKASI[Opens in a new window] and
STEPHEN GWYN ROBERTSOpen the ORCID record for STEPHEN GWYN ROBERTS[Opens in a new window]
SUDI MUNGKASI*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Sanata Dharma University, Yogyakarta, Indonesia email sudi@usd.ac.id
STEPHEN GWYN ROBERTS
Affiliation:
Mathematical Sciences Institute, College of Physical and Mathematical Sciences, Australian National University, Canberra, Australia email stephen.roberts@anu.edu.au
*
✉sudi@usd.ac.id
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Abstract

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The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.

Keywords

smoothness indicatorshock detectornumerical entropy productionfinite volume methodshallow water equations

MSC classification

Primary:35L65: Conservation laws
Secondary:65M08: Finite volume methods76M12: Finite volume methods

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 4 , October 2019, pp. 398 - 415
Copyright
© 2019 Australian Mathematical Society

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